Marta Morigi

Some structural results on the non-abelian tensor square of groups

Abstract We study the non-abelian tensor square G ⊗ G for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G/G′ so that G ⊗ G is isomorphic to the direct product of ∇(G) and the non-abelian exterior square G ∧ G. For any group G, we characterize the non-abelian exterior square G ∧ G in terms of a presentation of G. Finally, we apply our results to some classes of groups, such as the classes of free solvable and free nilpotent groups of finite rank, and some classes of finite p-groups.

Outer commutator words are uniformly concise

We prove that outer commutator words are uniformly concise, that is, if an outer commutator word u takes m different values in a group G, then the order of the verbal subgroup u(G) is bounded by a function depending only on m, and not on u or G. This is obtained as a consequence of a structure theorem for the subgroup u(G), which is valid if G is soluble, and without assuming that u takes finitely many values in G. More precisely, there is an abelian series of u(G), such that every section of the series can be generated by values of u all of whose powers are also values of u in that section. For the proof of this latter result, we introduce a new representation of outer commutator words by means of binary trees, and we use the structure of the trees to set up an appropriate induction.

Asymptotic Results for Transitive Permutation Groups

In this paper we give answers to some open questions concerning generation and enumeration of finite transitive permutation groups. In [1], Bryant, Kovács and Robinson proved that there is a number c′ such that each soluble transitive permutation group of degree n > 2 can be generated by [c′n/ √ log n ] elements, and later A. Lucchini [5] extended this result (with a different constant c′) to finite permutation groups containing a soluble transitive subgroup. We are now able to prove this theorem in full generality, and this solves the question of bounding the number of generators of a finite transitive permutation group in terms of its degree. The result obtained is the following.

Boundedly generated subgroups of finite groups

Abstract. The starting point for this work was the question whether every finite group G contains a two-generated subgroup H such that , where denotes the set of primes dividing the order of G. We answer the question in the affirmative and address the following more general problem. Let G be a finite group and let be a property of G. What is the minimum number t such that G contains a t-generated subgroup H satisfying the condition that ? In particular, we consider the situation where is the set of composition factors (up to isomorphism), the exponent, the prime graph, or the spectrum of the group G. We give a complete answer in the cases where is the prime graph or the spectrum (obtaining that in the former case and t can be arbitrarily large in the latter case). We also prove that if is the exponent of G, then t is at most four.

On the probability of generating free prosoluble groups of small rank

LetF be the free prosoluble group of rankd≤9. We study the minimum integerk such that the probability of generatingF withk elements is positive.

ON THE PROBABILITY OF GENERATING PROSOLUBLE GROUPS

LetF be the free prosoluble group of rankd. We determine the minimum integerk such that the probability of generatingF withk elements is positive.

A note on conciseness of Engel words

It is still an open problem to determine whether the nth Engel word [x, n y] is concise, that is, if for every group G such that the set of values e n (G) taken by [x, n y] on G is finite it follows that the verbal subgroup E n (G) generated by e n (G) is also finite. We prove that if e n (G) is finite, then [E n (G), G] is finite, and either G/[E n (G), G] is locally nilpotent and E n (G) is finite, or G has a finitely generated section that is an infinite simple n-Engel group. It follows that [x, n y] is concise if n is at most four.

GENERALIZING A THEOREM OF P. HALL ON FINITE-BY-NILPOTENT GROUPS

Let γ i (G) and Z i (G) denote the i-th terms of the lower and upper central series of a group G, respectively. In 1956 P. Hall showed that if γ i+1 (G) is finite, then the index |G: Z 2i (G)| is finite. We prove that the same result holds under the weaker hypothesis that |γ i+1 (G): γ i+1 (G) n Z i (G)| is finite.

Complements of the socle in monolithic groups

We show that if H is a finite group with a unique minimal normal subgroup N , which is not abelian, then the number of conjugacy classes of complements of N in H is strictly smaller than jN j.

Generating Permutation Groups

Abstract In this paper an algorithm is produced, which, given a permutation group G of degree n > 3, outputs a generating set for G with at most n/2 elements.